We note that \(p=((r-1)! \log n)^{2/(r(r-1))}\! / n^{2/r}\)  is the (sharp) threshold for the property that every vertex in \(\mathbb {G}(n,p)\)  is contained in an \(r\) -clique. Hence, thm:robust-corradi-hajnal is optimal up to a constant factor. That this is indeed the threshold for a \(K_r\) -factor in \(\mathbb {G}(n,p)\)  was first proved by Johansson, Kahn, and Vu [1]}. Furthermore, the threshold in the \(r = 3\)  case of thm:robust-corradi-hajnal was proved very recently by Allen, Böttcher, Corsten, Davies, Jenssen, Morris, Roberts, and Skokan [2]}. The counting statement in thm:robust-corradi-hajnal proves [2]} (which is new even for \(r = 3\) ).
